On codimension-one singular holomorphic foliations.

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Let F be a codimension one holomorphic foliation whose singular set Σ is contained in a compact leaf S of F.When F is of dimension one, Σ is a set of isolated points {q1, ..., qr}, C. Camacho and P. Sad define the index of F at each point qk and prove that the sum of these indices equals the Euler class c1(E) of the fibre bundle E normal to S.Generally, whenever Σ is of any dimension m, we can define a such index iα along the maximal dimension strates {Σα} of a suitable stratification of the complex variety Σ. Let σα be the fundamental cycle of Σα, σ the 2m-cycle of S defined by σ = Σiα.σα and σ* the 2-cocycle dual to σ by Poincaré isomorphism H2(S) → H2m(S), we prove that the cohomology class [σ*] equals the Euler class c1(E).